Question = Horse Vs Tower

Because the two sides start with four ranks between them, strong opening moves significantly increase your side's activity without having much impact on the opponent's, so I don't think there's as strong an incentive to leave things blocked as you suspect.

If someone is bored by the standard names of the chess pieces, I recommend to read this old thread.
lichess.org/forum/off-topic-discussion/a-more-politically-correct-way-of-identifying-your-pieces

I prefer:

King = The sorcerer
Queen = The witch
Rooks = The towers
Knights = The unicorns
Bishops = The twin brothers
Pawns = The eight dwarfs

Towers-rooks are usually stronger than Horsies-knights. Even in the opening, two rooks for one knight is a bad trade unless this leads to a quick mate.

@BrettKildahl I'll give a simpler example. If black made random moves, 1.e4 would be the best starter for white based on the logic that it opens up more squares for their pieces. The problem is that black can just answer 1.e5 and has already made progress towards solving all of his problems. So an example of a smarter move is 1.d4 where white can achieve e4 under more favorable circumstances where black gets no e5. This is an example of white making a concession, and not just being as active and energetic as possible with intent to get control instead.

In chess, what is good for one person is bad for the other. There are just two factions, white and black. I think chess should be played as trying to bargain for the most good and least bad, for your own pieces, relative to the most bad and the least good for theirs. I think this is clearly best. of course, there are incentives to have a little bit of contempt. Even when you can do so favorably, opening the game, causing direct confrontation between different parts of armies can lead to things like captures and recaptures bringing pieces off the board and bringing it closer to a resolution. Maybe there is some incentive to try and keep pieces on, to muddy the waters and create chances or outplay potential.

Your idea that rooks may never bridge the gap in value if the players are interested enough in hindering the opponent doesn't take into account the size of the board. Imagine a 5x8 board instead. Putting aside forced wins, you're dealing with a board on which you'd have a really strong case that rooks will never match knights in value. There isn't enough space. With four ranks between the players though, it's a different story.

Now imagine a 100x8 board on which pawns can move far enough for there to be the same relationship between an E pawn move and a D pawn move that you described. The rooks start off less mobile than the knights, but is there any doubt in your mind that they'll quickly overtake them in value?

Rook vs Knight is kind of like macro vs early aggression in RTS games. There is the question of Power now vs Power later and there are obvious conditions where you are inarguably wrong. For example, if one side lags hard in development, and has "Spectator pieces" then things like long term nutrition and macro goals go out the window because of a convertible / winning attack. Another situation, fully taking into account the size of the board is game states where black, not wanting to meet 1.d4 with it's mirror for fear of 2.c4 and white just being a leg up in a definitely slightly better position for obvious reasons, could try something weird like 1.f5 and putting all the pawns on light squares and looking for piece play on dark squares. Is it dubious and probably losing? Sure but good luck proving that unless you are very super human. This 1.f5 system would lag behind in development and you could have thematic battles like Can black be solid enough to catch up and then enjoy a rich and fertile game with many chances to get one or more of many potential and convoluted advantages, or does white get to open the position up and use development to roll over black? If it goes black's way, especially if there are no entry points, the rooks could be garbo for a very long and possibly indefinite time.

Perhaps I need to emphasize, that while we all (So far) seem to agree that opening the game and playing energetically is advantageous or desirable, it is only desirable on the premise of space and opening up path ways for the pieces. What if one player has more prospects to use it? I have a theory that opening up different channels will favor the person who has more threats and pieces in position to use them. I think that person will usually be white because they get an extra half move, of course. In fact, I think a lot of black's chances (according to my own self learned and invented way for how I think optimal chess should be played) revolves around dampening or stealing that move advantage from the first player!

oh hey, bishops are the strongest and towers are stronger than unicorns

calculated statistics from 100k games, only kill count for now:
docs.google.com/spreadsheets/d/1wbW7YdxtiIdHMax1mCM6nB5sVSWUeaUCsH7inMu1nTQ/edit?usp=sharing

I'll add more statistics later, let me know if you have any ideas.

@Saranskich is there a way to calculate this according to different phases of the game and different game states? Bishops are weird because they are likely to be used early and survive a very long time depending on how both sides play so they might have more game time to wrack up points.